## Is this a rigorous proof?

It appears simple but I am concerned about the rates that each function approaches $\infty$. Should I be?

Given:

$1+R=1+(T-t)L(t,T)$

and

$1+R=e^{(T-t)y(t,T)$

show that

$\lim_{T \rightarrow t}L(t,T) = \lim_{T \rightarrow t}y(t,T)$.

PROOF:

Rearrange both formulae to get:

$L(t,T)=\frac{R}{(T-t)}$

and

$y(t,T)=\frac{ln(1+R)}{(T-t)}$

Take the limit of each as [tex] T \rightarrow t:

$\lim_{T \rightarrow t}L(t,T)=\lim_{T \rightarrow t}\frac{R}{(T-t)} = \infty$

and

$\lim_{T \rightarrow t}y(t,T)=\lim_{T \rightarrow t}\frac{ln(1+R)}{(T-t)} = \infty$

As both approach $\infty$ we can state that:

$\lim_{T \rightarrow t}L(t,T) = \lim_{T \rightarrow t}y(t,T)$.